====== Michael & Slava's invasion plans ======

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====== The Question ======

Michael recently presented some current work at a MathBio "chalk talk"((on 2012-01-26)).  He and Slava are looking at invasion of a rare clonal mutant in the presence of a resident population.  He presented the following equation as a starting point:
<div right><BOOKMARK:eq_1>(Eq. 1)</div>
\[
  \diff{n(x)}{t} = r n(x) \left[ 1 - \frac{ n(x) + \alpha(x,y) n(y) }{K(x)} \right]
\]
where \(x,y\) represent the two phenotypes and the strength of competition with one's own type is normalized, \(\alpha(x,x)=1\).  The equation is modeled after the logistic equation except each phenotype competes both with its own type and with the other.

They then go on to make some simplifying assumptions.  First, they assume a monomorphic resident population.  If \(x\) represents the residents then the solution is given by \(n(x)=K(x), n(y)=0\).  Next they ask what are the conditions that will allow a mutant \(y\) to invade, assuming the mutant is rare, ie. \(n(y) \ll \alpha(y,x) n(x)\) and \(n(x)\approx K(x)\)?  They derive the following equation for the mutant dynamics:
<div right><BOOKMARK:eq_2>(Eq. 2)</div>
\[
  \diff{n(y)}{t} = r n(y) \left[ 1 - \frac{ \alpha(y,x) K(x) }{K(y)} \right].
\]

Given my obsession with reaction kinetics I wondered what is the simplest reaction set that can produce the above equation?

====== Simple birth and competitive death ======

My first guess was
\[
\begin{array}{rcll}
  N(u)   & \xrightarrow{\beta(u)}          & 2 N(u)     & \text{simple birth} \\
  N(u) + N(v) & \xrightarrow{\omega(u,v)}  & N(v)       & \text{competitive death}
\end{array}
\]
where \((u,v)\) represent any combination of \((x,y)\).

The resident dynamics (neglecting mutants) is
\[
  \diff{n(x)}{t} = n(x) [ \beta(x) - \omega(x,x) n(x) ]
\]
so the residents are in equilibrium at \(n(x) = K(x) \equiv \beta(x)/\omega(x,x)\).

The mutants, when rare, would then follow
<div right><BOOKMARK:eq_3>(Eq. 3)</div>
\[
  \diff{n(y)}{t} = n(y) [ \beta(y) - \omega(y,x) K(x) ].
\]

So this can work.  To match [[#Eq. 2]] and [[#Eq. 3]] we require 
\[
\begin{array}{rcl}
  \beta(x) & \equiv & r \\
  \omega(y,x) & \equiv & r \frac{\alpha(y,x)}{K(y)}.
\end{array}
\]
Interestingly, we require that the birth rate be independent of phenotype and death ties together both competitive \(\alpha\) and carrying capacity \(K\) effects.((Also, recall \(\alpha(x,x)=1\) so we satisfy the consistency condition that \(\omega(x,x)=\omega(y\rightarrow x,x)\).))

Note that this approach gives the same invasion dynamics ([[#Eq. 2]]) and the same full model ([[#Eq. 1]]), ie. if we don't assume residents in equilibrium or mutants rare.


====== Competitive birth and simple death ======

I wondered if other possibilities were also valid.
\[
\begin{array}{rcll}
  E(u) + N(v) & \xrightarrow{\gamma(u,v)}  & N(u) + N(v) & \text{competitive birth} \\ 
  N(u)   & \xrightarrow{\delta(u)}         & E(u)   & \text{simple death}
\end{array}
\]
where \(E\) represents an "empty" (available) state and again \((u,v)\) represent any combination of \((x,y)\).

The resident dynamics (neglecting mutants) is
\[
  \diff{n(x)}{t} = \gamma(x,x) e(x) n(x) - \delta(x) n(x) = - \diff{e(x)}{t}.
\]
Since \(n(x)+e(x)\equiv Q(x)\) is constant over time we can write
\[
  \diff{n(x)}{t} = n(x) [ \gamma(x,x)Q(x)-\delta(x) - \gamma(x,x) n(x) ]
\]
so the residents are in equilibrium at \(n(x) = K(x) \equiv Q(x) - \delta(x)/\gamma(x,x)\).

The mutants, when rare, would then follow
<div right><BOOKMARK:eq_4>(Eq. 4)</div>
\[
\begin{array}{rcl}
  \diff{n(y)}{t} & = & \gamma(y,x) e(y) n(x) - \delta(y) n(y) \\
    & = & \gamma(y,x) [Q(y) - n(y)] n(x) - \delta(y) n(y) \\
    & = & \gamma(y,x) Q(y) n(x) - n(y) [ \gamma(y,x) n(x) - \delta(y) ].
\end{array}
\]

The leading term needs to be null to match [[#Eq. 2]] but that can't be except in the trivial cases \(\gamma(y,x)=0\) (no births allowed at \(y\)) or \(Q(y)=0\) (\(y\) is barren).  So these reactions cannot produce the necessary dynamics.

On reflection I see why: we assume mutations are negligibly rare so we don't want production of new mutants to occur through our reactions.  So, births need to be a local process -- residents at \(x\) can't produce mutants at \(y\)((Remember, we're talking about //phenotype// space.)).

====== Competitive birth and death ======

If we need births to be local then deaths need to be "non-local", ie. depend on either sub-population.  I think we can do that if both birth and death are competitive processes.
\[
\begin{array}{rcll}
  E(u) + N(u) & \xrightarrow{\gamma(u)}    & N(u) + N(v) & \text{local competitive birth} \\ 
  N(u) + N(v) & \xrightarrow{\omega(u,v)}  & E(u) + N(v) & \text{competetive death}
\end{array}
\]
where \(E\) represents an "empty" (available) state and again \((u,v)\) represent any combination of \((x,y)\).

The resident dynamics (neglecting mutants) is
\[
  \diff{n(x)}{t} = \gamma(x) e(x) n(x) - \omega(x) n(x)^2 = - \diff{e(x)}{t}.
\]
Since \(n(x)+e(x)\equiv Q(x)\) is constant over time we can write
\[
  \diff{n(x)}{t} = n(x) \left[ \gamma(x)Q(x) - n(x) \left( \gamma(x) + \omega(x,x) \right) \right]
\]
so the residents are in equilibrium at 
\[
  n(x) = K(x) \equiv Q(x) \frac{\gamma(x)}{\gamma(x) + \omega(x,x)}.
\]

The mutants, when rare, would then follow
<div right><BOOKMARK:eq_5>(Eq. 5)</div>
\[
\begin{array}{rcl}
  \diff{n(y)}{t} & = & \gamma(y) e(y) n(y) - \omega(y,x) n(y) n(x) \\
    & = & n(y) \left[ \gamma(y) Q(y) - \left( \gamma(y) n(y) + \omega(y,x) n(x) \right) \right] \\
    & \approx & \gamma(y) Q(y) n(y) \left[ 1 - \frac{\omega(y,x) K(x)}{\gamma(y) Q(y)} \right].
\end{array}
\]

To match [[#Eq. 2]] we need
\[
\begin{array}{rcl}
  Q(y) & = & r/\gamma(y) \\
  \gamma(x) & = & r/K(x) - \omega(x,x) \\
  \omega(y,x) & = & r \frac{\alpha (y,x)}{K(y)}
\end{array}
\]
or, strictly in terms of the original functions,
\[
\begin{array}{rcl}
  \omega(y,x) & = & r \frac{\alpha (y,x)}{K(y)} \\
  \gamma(x) & = & \frac{r}{K(x)} [ 1 - \alpha(x,x) ] = 0 \\
  Q(y) & = & K(y)/ [ 1 - \alpha(y,y) ] = \infty.
\end{array}
\]
So this is a bit weird: zero birthrate but infinite empty (available) states.

====== Conclusions ======

It looks like the best answer is the simplest: use [[#Simple birth and competitive death]] for the individual-based representation of Michael and Slava's population-level model.
